Lévy and Vaught showed that inaccessible cardinals can be indexed only relative to a model; across models, the same cardinal may occupy different finite positions in the hierarchy.

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Across models(comparing different logical systems): When comparing between different mathematical systems or frameworks rather than staying within just one.
Inaccessible cardinals(as used in philosophy of mathematics): A technical concept from mathematical logic referring to very large numbers with special properties; used in abstract mathematics but doesn't correspond to anything we can physically point to or construct.
Indexed(used to describe how cardinals are numbered or positioned): Assigned a number or position in an ordered list, like how pages in a book are numbered to show their order.
Lévy and Vaught(named researchers in the field): Two mathematicians and logicians (Azriel Lévy and Robert Vaught) who made important discoveries about how certain kinds of numbers work in mathematical logic.

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